Complex analysis - electron screening

[kknull]

Hi!
I have to understand how this integral is evaluated (it is taken from Fetter - Quantum theory of many particle systems)(14.24):

http://dl.dropbox.com/u/158338/fis/fetter.pdf"

in particular, i don't know how the log brach cuts are defined..
as far as I know, log branch cuts can be define as the regions at fixed log argument.. (i.e. -inf to 0 for [-pi;pi[).
Gnuplot says that the "natural" branch cut ([-pi;pi[) is the discontiunous line in this pic:

http://dl.dropbox.com/u/158338/fis/arg-rminus.png"

ideas?
thanks!

 

[kknull]

sorry, the right pdf link is http://dl.dropbox.com/u/158338/fis/fetter2.pdf"

also, I can't understand how the integral contour is chosen.

thanks :)

 

[rsq_a]

sorry, the right pdf link is http://dl.dropbox.com/u/158338/fis/fetter2.pdf"

also, I can't understand how the integral contour is chosen.

thanks :)

I had a quick look. The log branch cuts are obviously taken directly upwards (as in the figure). They are using Cauchy's Theorem applied to a contour which goes from -infinity to +infinity along the real axis (the original contour for the integral), plus a semi-circle into the upper half plane. However, since you can't really pass through the cuts, you have to slit the semi-circle along the cuts.

By Cauchy's Theorem, that is equal to the residue at the pole along the imaginary axis.

 

[kknull]

ok,
so is it a sort of jordan lemma? the integral along the semi circle is not 0, but it's the integral along C1 and C2.
In other words:

integral + C1 + C2 = residue of the pole
integral = residue of the pole - C1 - C2

is it correct?

thanks